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2 years ago

45 marbles in 9 bags; 150 marbles in 36 bags is this rate proportional?

Mathematics

1 answer:

stellarik [79]2 years ago

7 0

Answer:

No, the rate is not proportional.

Step-by-step explanation:

If there are 45 marbles in 9 bags, we can assume that there are 5 marbles in each bag.

Going by that rate, we can do 5 * 36 = 180, NOT 150.

Thus, the rate is not proportional.

Another way to solve would be to do 45/9 and 150/36. If the rate is proportional, the answer for both should be the same.

However, 45/9=5 and 150/36=4.16666666667.

Those two answers are NOT equal.

Thus, the rate is not proportional.

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Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $\bullet$ $f(x)$ is divisible by $x^3$. $\bullet

frosja888 [35]

There seems to be one character missing. But I gather that f(x) needs to satisfy

• $x^3$ divides $f(x)$

• $(x-1)^3$ divides $f(x)^2$

I'll also assume f(x) is monic, meaning the coefficient of the leading term is 1, or

$f(x) = x^5 + \cdots$

Since $x^3$ divides $f(x)$ , and

$f(x) = x^3 p(x)$

where $p(x)$ is degree-2, and we can write it as

$f(x)=x^3 (x^2+ax+b)$

Now, we have

$f(x)^2 = \left(x^3p(x)\right)^2 = x^6 p(x)^2$

so if $(x - 1)^3$ divides $f(x)^2$ , then $p(x)$ is degree-2, so $p(x)^2$ is degree-4, and we can write

$p(x)^2 = (x-1)^3 q(x)$

where $q(x)$ is degree-1.

Expanding the left side gives

$p(x)^2 = x^4 + 2ax^3 + (a^2+2b)x^2 + 2abx + b^2$

and dividing by $(x-1)^3$ leaves no remainder. If we actually compute the quotient, we wind up with

$\dfrac{p(x)^2}{(x-1)^3} = \underbrace{x + 2a + 3}_{q(x)} + \dfrac{(a^2+6a+2b+6)x^2 + (2ab-6a-8)x +2a+b^2+3}{(x-1)^3}$

If the remainder is supposed to be zero, then

$\begin{cases}a^2+6a+2b+6 = 0 \\ 2ab-6a-8 = 0 \\ 2a+b^2+3 = 0\end{cases}$

Adding these equations together and grouping terms, we get

$(a^2+2ab+b^2) + (2a+2b) + (6-8+3) = 0 \\\\ (a+b)^2 + 2(a+b) + 1 = 0 \\\\ (a+b+1)^2 = 0 \implies a+b = -1$

Then $b=-1-a$ , and you can solve for a and b by substituting this into any of the three equations above. For instance,

$2a+(-1-a)^2 + 3 = 0 \\\\ a^2 + 4a + 4 = 0 \\\\ (a+2)^2 = 0 \implies a=-2 \implies b=1$

So, we end up with

$p(x) = x^2 - 2x + 1 \\\\ \implies f(x) = x^3 (x^2 - 2x + 1) = \boxed{x^5-2x^4+x^3}$

6 0

2 years ago

Katy invests a total of $26,500 in two accounts paying 4% and 9% annual interest, respectively. How much was invested in each ac

White raven [17]

A = amount invested at 4%.

b = amount invested at 9%.

we know the total amount invested was $26500, thus a + b = 26500.

whatever% of anything is just (whatever/100) * anything.

how much is 4% of a? well, is just (4/100) * a, or 0.04a.

how much is 9% of b? well, is just (9/100) * b, or 0.09b.

we know the interest yielded for both amounts adds up to $1510, thus 0.04a + 0.09b = 1510.

$\bf \begin{cases}
a+b=26500\implies \boxed{b}=26500-a\\
0.04a+0.09b=1510\\
----------------\\
0.04a+0.09\left( \boxed{26500-a} \right)=1510
\end{cases}
\\\\\\
0.04-0.09a+2385=1510\implies -0.05a=-875
\\\\\\
a=\cfrac{-875}{-0.05}\implies a=17500$

how much was invested at 9%? well, b = 26500 - a.

7 0

2 years ago

Write V – 250 in simplest radical form.

BartSMP [9]

Answer:

5i√10

Step-by-step explanation:

hope this answers your question

3 0

2 years ago

Read 2 more answers

Will markk the brainliest

ivolga24 [154]

Answer:

1. the range of f^-1(x) is {10, 20, 30}.

2. the graph of f^-1(x) will include the point (0, 3)

3. n = 8

Step-by-step explanation:

1. The domain of a function is the range of its inverse, and vice versa. The range of f^-1(x) is {10, 20, 30}.

2. See above. The domain and range are swapped between a function and its inverse. That means function point (3, 0) will correspond to inverse function point (0, 3).

3. The n-th term of an arithmetic sequence is given by ...

an = a1 +d(n -1)

You are given a1 = 2, a12 = 211, so ...

211 = 2 + d(12 -1)

209/11 = d = 19 . . . . . solve the above equation for the common difference

Now, we can use the same equation to find n for an = 135.

135 = 2 + 19(n -1)

133/19 = n -1 . . . . . . . subtract 2, divide by 19

7 +1 = n = 8 . . . . . . . . add 1

135 is the 8th term of the sequence.

3 0

2 years ago

O A. 17 degrees O B. 163 degrees O C. 73 degrees

eduard

Answer: B 163 degrees

Step-by-step explanation:

That's an obtuse angle, bigger than 90 degrees, so we read the appropriate upper scale for the angle.

6 0

2 years ago

Read 2 more answers